* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            sqr(x) -> *(x,x)
            sum(0()) -> 0()
            sum(s(x)) -> +(*(s(x),s(x)),sum(x))
            sum(s(x)) -> +(sqr(s(x)),sum(x))
        - Signature:
            {sqr/1,sum/1} / {*/2,+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sqr,sum} and constructors {*,+,0,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            sqr(x) -> *(x,x)
            sum(0()) -> 0()
            sum(s(x)) -> +(*(s(x),s(x)),sum(x))
            sum(s(x)) -> +(sqr(s(x)),sum(x))
        - Signature:
            {sqr/1,sum/1} / {*/2,+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sqr,sum} and constructors {*,+,0,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          sum(x){x -> s(x)} =
            sum(s(x)) ->^+ +(*(s(x),s(x)),sum(x))
              = C[sum(x) = sum(x){}]

** Step 1.b:1: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            sqr(x) -> *(x,x)
            sum(0()) -> 0()
            sum(s(x)) -> +(*(s(x),s(x)),sum(x))
            sum(s(x)) -> +(sqr(s(x)),sum(x))
        - Signature:
            {sqr/1,sum/1} / {*/2,+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sqr,sum} and constructors {*,+,0,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(+) = {1,2}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
              p(*) = [8]                  
              p(+) = [1] x1 + [1] x2 + [2]
              p(0) = [3]                  
              p(s) = [1] x1 + [5]         
            p(sqr) = [8]                  
            p(sum) = [3] x1 + [6]         
          
          Following rules are strictly oriented:
           sum(0()) = [15]                  
                    > [3]                   
                    = 0()                   
          
          sum(s(x)) = [3] x + [21]          
                    > [3] x + [16]          
                    = +(*(s(x),s(x)),sum(x))
          
          sum(s(x)) = [3] x + [21]          
                    > [3] x + [16]          
                    = +(sqr(s(x)),sum(x))   
          
          
          Following rules are (at-least) weakly oriented:
          sqr(x) =  [8]   
                 >= [8]   
                 =  *(x,x)
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:2: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            sqr(x) -> *(x,x)
        - Weak TRS:
            sum(0()) -> 0()
            sum(s(x)) -> +(*(s(x),s(x)),sum(x))
            sum(s(x)) -> +(sqr(s(x)),sum(x))
        - Signature:
            {sqr/1,sum/1} / {*/2,+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sqr,sum} and constructors {*,+,0,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(+) = {1,2}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
              p(*) = [0]                  
              p(+) = [1] x1 + [1] x2 + [0]
              p(0) = [0]                  
              p(s) = [1] x1 + [1]         
            p(sqr) = [1]                  
            p(sum) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
          sqr(x) = [1]   
                 > [0]   
                 = *(x,x)
          
          
          Following rules are (at-least) weakly oriented:
           sum(0()) =  [0]                   
                    >= [0]                   
                    =  0()                   
          
          sum(s(x)) =  [1] x + [1]           
                    >= [1] x + [0]           
                    =  +(*(s(x),s(x)),sum(x))
          
          sum(s(x)) =  [1] x + [1]           
                    >= [1] x + [1]           
                    =  +(sqr(s(x)),sum(x))   
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            sqr(x) -> *(x,x)
            sum(0()) -> 0()
            sum(s(x)) -> +(*(s(x),s(x)),sum(x))
            sum(s(x)) -> +(sqr(s(x)),sum(x))
        - Signature:
            {sqr/1,sum/1} / {*/2,+/2,0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {sqr,sum} and constructors {*,+,0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^1))